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. 5N*f
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.9 6 = E =()
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P"q%e= - FStrg='e?lq
ciP z w
s a = -
MAE 261A
E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS
FALL 2004
Homework Assignment No. 3
Due Wednesday, November 3, 2004.
Problem 1
Take the first variation of the functionals listed below, derive the Euler-Lagrange equations, a
MAE 261A
E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS
FALL 2004
Homework Assignment No. 4
Not to be collected or graded.
Problem 1
Consider the minimization problem
b
min
a
F (x, y, y )dx + f (z)
subject to:
b
G(x, y, y )dx +
M* e-b l ft J-e 9'> L-?to,.-t 4 a*ew o-.lz /+s siXn,r<,a*Prolzk* |
Fncc
zsoz)
! " . j
-.t"fG"f
c .9 E
t a
,t-
I r 2c.'t,u /^')2-z^lJY
-z -rzt/ S*" t 2tn"8u' 'r 2tatf,6a/ S^l )x I ft f ,^" s*"
t f'['o-t-'zu"87
+ftl ^*t
* = 'i"su -=-u"Eu-z8^\
MAE 261A
E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS
FALL 2004
Homework Assignment No. 1 Solutions
Due Wednesday, October 13, 2004.
Problem 1
Let u, v, and w be vectors. Using index notation (and the Einstein summation conventio
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 6: Castiglianos Theorem, Hamiltons Principle, Ritz
Method, and Calculus of Variations
Assigned Tuesday, October 26, 2010.
Due Thursday, November 4, 2010 by 7:59
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 5: Castiglianos Theorem
Assigned Wednesday, October 20, 2010.
Due Thursday, October 28, 2010 by 8:00 am.
Derive the complementary strain energy for the Timoshen
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 4: Derivation of Structural Theories and Energy Theorems
Assigned Wednesday, October 13, 2010.
Due Thursday, October 21, 2010 by 8:00 am.
Problem I. Timoshenko
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 3: Calculus of Variations and Variational Principles
Assigned Wednesday, October 6, 2010.
Due Thursday, October 14, 2010 by 8:00 am.
Problem I. Equations of Ela
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 2: Overview Field Equations of Elasticity
Assigned Wednesday, September 29, 2010.
Due Thursday, October 7th, 2010 by 8:00 am.
Study Lecture #1 - #2 materials al
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 1: Overview Field Equations of Elasticity
Assigned Wednesday, September 22, 2010.
Due Thursday, September 30, 2010 by 8:00 am.
Study Lecture #1 - #2 materials a
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2009
Homework Set 0: Indicial Notation
Assigned Tuesday, September 14, 2010.
Due Tuesday, September 28, 2010 by 8:00 am.
Study Lecture # 0 materials along with Appendix I of Ener
UCLA Mechanical & Aerospace Engineering Department
MAE 261A: E NERGY AND C OMPUTATIONAL M ETHODS IN
S TRUCTURAL M ECHANICS
Fall 2013
Lecture: T, Th, 4:00pm5:50pm, BOELTER 5419
Instructor: Dr. Vinay K. Goyal (vinaykgoyal@gmail.com)
Ofce Hours: T, Th: 3:45-
cients ci s did not have physical meaning, while unknown coefcients w1 , w2 , 1 , 2 in the
new form have a clear physical meaning. Here w1 and w2 represent the end transverse displacements and 1 and 2 represents the end rotations.
M AE 261A Substitute the
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 8: Structural Applications
Assigned Tuesday, November 8, 2010.
Due Thursday, November 18, 2010 by 7:59 am.
1. Solve Problem 5.6 of the textbook. State the assum
MAE 256A
Linear Elasticity
Instructor: Professor Ajit Mal, 310-825-5481 ajit@ucla.edu
University of California, Los Angeles
School of Engineering and Applied Science
Mechanical and Aerospace Engineering (MAE) Department
Text: Mechanical & Aerospace Engine
MAE 256B, HW #3
Course Reader Problems: 2.1, 2.4, 3.6, 3.7, 3.8, 3.9
Hint for Problem 3.6
Assume that the traction free plane has the normal n with components ni. Since the plane
through the point has no traction:
t i ijn j 0
Write down the three componen
MSOL MAE 256A
HW # 1
Problem 1. Determine whether the vectors
( )
(
),
( )
( )
( ),
(
)
are linearly independent or linearly dependent. If they are linearly dependent, find a linear relation
among them.
Problem 2. Solve the following system of equations o
CES 4141 - Stress Analysis
Summer 1998
Direct Stiffness - Truss Application
The following notes present the direct stiffness method for matrix truss analysis. We will generalize to include beam elements later.
Symbols: Well stay consistent with the notati
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
FALL 2010
Homework Set 9: Finite Element Formulation
Assigned Wednesday, November 17, 2010.
Due Tuesday, November 30, 2010.
1. Develop the nite element formulation for Timoshenko beam
Concrete Pavement Guide Part 2: New Construction
Division of Maintenance Pavement Program
CHAPTER 200 Continuously Reinforced Concrete Pavement (CRCP)
January 2015
Disclaimer
The contents of this guide reflect the views of the authors who are responsible
MAE 256A
Linear Elasticity
2008 Ajit Mal
Instructor: Professor Ajit Mal, 310-825-5481 ajit@ucla.edu
University of California, Los Angeles
School of Engineering and Applied Science
Mechanical and Aerospace Engineering (MAE) Department
Text: Mechanical & Ae
Human Sexuality
Exam #2 Study Guide
Chapter 5: Gender & Gender Roles
Know about gender, gender roles (stereotypes/socialization), and gender identity
Know about transgender, gender dysphoria, and transsexual
What is cognitive social learning theory
What i
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
Derivation of Structural Theories
Problem I. Timoshenko Beam Theory Using Principle of Virtual Work The purpose of this problem
is to derive Timoshenko beam theory. The theory account
Figure 2: Rigid bar and wires from Problem 2.
M AE 261A
Problem 3
E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS
Consider the pin-connected structure shown in Figure 3. Suppose that the material of all members obeys the following
stress-str
M AE 261A
E NERGY AND C OMPUTATIONAL M ETHODS IN S TRUCTURAL M ECHANICS
Applied Calculus of Variations
1. To derive the principle of virtual work, the displacement eld is varied while keeping the external
forces xed. In this problem the tractions and the
MAE 261A
E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS
FALL 2004
Homework Assignment No. 1
Due Wednesday, October 13, 2004.
Problem 1
Let u, v, and w be vectors. Using index notation (and the Einstein summation convention), prove
UCLA M ECHANICAL & A EROSPACE E NGINEERING D EPARTMENT
MAE 261A- E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS FALL 2004
Lecture: M&W, 8:00am9:50am, 5436 Boelter Hall. Instructor: Professor William S. Klug 48-121B E. IV klug@seas.uc
MAE 261A
E NERGY AND VARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS
FALL 2004
Homework Assignment No. 7
Due Friday, December 10, 2004.
Problem 1
Examine the following problem for self-adjointness and positive definiteness using the normal inne